There are two problems for nonconvex functions under the weak Wolfe–Powell line search in unconstrained optimization problems. The first one is the global convergence of the Polak–Ribiere–Polyak conjugate gradient algorithm and the second is the global convergence of the Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method. Many scholars have proven that the two problems do not converge, even under an exact line search. Two circle counterexamples were proposed to generate the nonconvergence of the Polak–Ribiere–Polyak algorithm for the nonconvex functions under the exact line search, which inspired us to define a new technique to jump out of the circle point and obtain the global convergence. Thus, a new Polak–Ribiere–Polyak algorithm is designed by the following steps. (i) Given the current point and a parabolic surface is designed; (ii) An assistant point is defined based on the current point; (iii) The assistant point is projected onto the surface to generate the next point; (iv) The presented algorithm has the global convergence for nonconvex functions with the weak Wolfe–Powell line search. A similar technique is used for the quasi-Newton method to get a new quasi-Newton algorithm and to establish its global convergence. Numerical results show that the given algorithms are more competitive than other similar algorithms. Meanwhile, the well-known hydrologic engineering application problem, called parameter estimation problem of nonlinear Muskingum model, is also done by the proposed algorithms.

中文翻译：

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无约束优化问题中两个开放问题的投影技术

在无约束优化问题中，弱 Wolfe-Powell 线搜索下的非凸函数有两个问题。第一个是Polak-Ribiere-Polyak共轭梯度算法的全局收敛，第二个是Broyden-Fletcher-Goldfarb-Shanno拟牛顿法的全局收敛。许多学者已经证明，即使在精确的线搜索下，这两个问题也不收敛。提出了两个圆反例来生成精确线搜索下非凸函数的Polak-Ribiere-Polyak算法的不收敛性，这启发我们定义了一种跳出圆点并获得全局收敛的新技术。因此，通过以下步骤设计了一种新的 Polak-Ribiere-Polyak 算法。(i) 给定当前点并设计一个抛物面；(ii) 在当前点的基础上定义一个辅助点；(iii) 将辅助点投影到曲面上生成下一个点；(iv) 所提出的算法具有非凸函数与弱 Wolfe-Powell 线搜索的全局收敛性。类似的技术被用于拟牛顿法以获得新的拟牛顿算法并建立其全局收敛性。数值结果表明，给定的算法比其他同类算法更具竞争力。同时，著名的水文工程应用问题，称为非线性Muskingum模型的参数估计问题，也由所提出的算法完成。(iv) 所提出的算法具有非凸函数与弱 Wolfe-Powell 线搜索的全局收敛性。类似的技术被用于拟牛顿法以获得新的拟牛顿算法并建立其全局收敛性。数值结果表明，给定的算法比其他同类算法更具竞争力。同时，著名的水文工程应用问题，称为非线性Muskingum模型的参数估计问题，也由所提出的算法完成。(iv) 所提出的算法具有非凸函数与弱 Wolfe-Powell 线搜索的全局收敛性。类似的技术被用于拟牛顿法以获得新的拟牛顿算法并建立其全局收敛性。数值结果表明，给定的算法比其他同类算法更具竞争力。同时，著名的水文工程应用问题，称为非线性Muskingum模型的参数估计问题，也由所提出的算法完成。